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Chapter 2

LITERATURE REVIEW

___________________________________________________________

A thorough literature survey is carried out in the present

chapter on the tolerance allocation problem of mechanical assemblies.

The survey includes both traditional as well as evolutionary

algorithms. Based on the literature review, the gaps have been

identified. Finally, this chapter has defined the aims and objective of

the present thesis.

The aim of tolerance allocation for any component in an

assembly is to have interchangeability. However, the tolerance value

that is assigned to any dimension of the component should not affect

its functionality. The close tolerance leads to high manufacturing cost,

whereas loose tolerance leads to the improper functionality. Some of

the existing literature that studied the impact of tolerance allocation of

mechanical assembly is mentioned below.

Two different Tolerancing methods were addressed by Evan’s

(Evans 1974, 1975). They are statistical and worst-case methods.

The authors had explained the activities that are to be carried out on

designing the tolerances. Recently, many researchers are following the

similar pattern of Tolerancing approaches developed by him. Later on,

Nigam and Tuner (1995) published another review on the same

research area. Only few changes involving minor improvements were

observed in those two decades.

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The analysis of tolerances related to one-dimensional and

2D/3D assemblies with kinematic adjustments were presented by

Chase and Greenwood (1988) and Chase and Parkinson (1991),

respectively. Moreover, Gerth (1996), Kumar et. al (1992) and Wu et.

al (1988) given different formulations for tolerance stack-up and cost-

tolerance analysis. Ngoi and Ong (1998) were made little

improvements in the same area. Later on, the comparison of usage of

different Tolerancing formulae in tolerance allocation problems was

given by Graves (1999).

Voelcker (1993) introduced the concept of geometric and

parametric tolerancing schemes, with more importance on the

metrology and later on given emphasis for tolerancing on assembly

(1998). Inspection data analysis was conducted only for the tolerance

evaluation by Feng and Hopp (1991). Ngoi and Kuan (1995) reviewed

one-dimensional +/- tolerance charting, which is critical for tolerance

exchange.

2.1 TOLERANCING SCHEMES

Last two decades had witnessed enormous development in the

tolerance allocation of mechanical assemblies without influencing the

cost and functionality. There are two types of tolerancing schemes:

parametric and geometric Tolerancing schemes.

The process of recognizing a set of parameters and allocating

limits to those parameters which defines the span of values is termed

as parametric Tolerancing and was developed by Requicha (1993).

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Conventional +/- Tolerancing comes under this category. Vectorial

Tolerancing is also a kind of parametric Tolerancing and was

introduced by Wirtz (1991).

The tolerance values of the features, like orientation, forms,

run-outs, location and profiles, are allocated using Geometric

Tolerancing. Even though, the draw backs of parametric Tolerancing

is addressed by geometric Tolerancing, it still posses some draw

backs.

Statistical Tolerancing is an advancement of classical

parametric Tolerancing, which is a substitute to worst-case

Tolerancing. However, there is no established statistical explanation

for the geometric tolerances (Voelcker 1995). Similarly, the present

standard ASME Y4.15M-1994 also represents statistical tolerancing

as a variation of +/- tolerancing. This point was clearly explained by

Srinivasan (1994b). The same author worked on formulating the

statistical tolerance zones after using systematic procedures (1997).

Moreover, Wuz (1997) used a Rayleigh distribution to model the

positional deviation. The sensitivity examination for geometric

tolerance was introduced by the same author.

2.2 TOLERANCE ANALYSIS AND SYNTHESIS

Tolerance Analysis is a technique of verifying the functionality of

a design, after considering the variability of different parts. Tolerance

synthesis is a method of assigning tolerances to individuals based on

the functionality. Majority of the techniques published were based on

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using the optimization strategy for cost-tolerance objective function.

They usually work by setting up the tolerance ‘types’ and try to obtain

the optimal tolerance ‘values’. Later on, researches developed various

dimensional tolerance chain models which are discussed below.

Linear tolerance stack-up models were very old Tolerancing

technique. The first book that described the traditional tolerance

stack-up conditions was written by Fortini (1967). The most widely

used tolerance accumulation models, namely worst case and

statistical method of root sum square were elaborately discussed in it.

The equations representing the worst case and RSS are given as

n

1iiASM TT and

n

1i

2iASM TT , respectively. Then, Greenwood and

Chase (1987) developed a new tolerance stack-up condition called

estimated mean shift (EMS) model with the knowledge of worst case

and root sum square methods. They also applied the above models to

nonlinear problems (Greenwood and Chase 1988, 1990). Later on,

some researchers from the same group published similar type of

analysis results for 2D and 3D assemblies (Chase et. al. 1995, 1996,

1997, 1998, Gao et. al. 1998a, b). On the same guide lines, Zhang and

Wang (1993) analyzed the tolerances for a cam mechanism. Moreover,

a macroscopic construction for tolerance analysis and allocation was

established by Zhang (1996) and Zhang and Porchet (1993). This work

integrates both the design and manufacturing tolerances to achieve

simultaneous tolerancing.

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Bjorke (1989) had derived different cases of gaps and spans in

detail based on the chain links. Later on, Bjorke’s work was extended

by Treacy et. al (1991) for establishing the automatic tolerance chain

generation. Moreover, the finite range probability density function

using numerical convolution was developed by Varghese et. al (1996).

Skowronski and Turner (1997) used a method of variance reduction in

the Monte Carlo tolerance analysis, which is used in commercial soft-

wares, like variation simulation analysis software. In addition to these

methods, Hutchings (1999) and Roy et. al. (1991) developed software

for tolerance analysis. Tolerance analysis using dimensional tolerance

chain models were developed by Ligget (1993), Kirschling (1991) and

Spotts (1983).

Ngoi and Ong (1998) developed a Tolerance synthesis method

for allocating assembly functional tolerances to each part tolerance.

Based on tolerance-cost models, many tolerance allocation problems

are optimized. In most of the methods, due to the inherent difficulty of

optimization strategies, only dimensional tolerances were considered.

Sutherland and Roth (1975), Spotts (1973) and Speckhart (1972) had

given initial efforts to optimize the tolerance allocation problem by

considering the minimization of manufacturing cost as objective

function. By utilizing the production tolerance-cost data curve, the

authors developed reciprocal power model, reciprocal squared cost

model and an exponential cost model, respectively. The above

research concentrated on developing an analytical model for tolerance-

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cost relations, and optimization of corresponding tolerances. Wilde

(1975) utilized pseudo-boolean programming for the formulation of

tolerance allocation problem.

To incorporate the allocation of optimal manufacturing

tolerances Michael and Siddall (1981, 1982) enhanced the traditional

design optimization problem. This method helped in creating an

optimal area of interest rather than a single point. The draw back

associated with this approach is that it is computationally more

expensive. Parkinson (1982, 1984, 1985) as an approximation applied

reliability analysis to estimate the probability of failure.

The tolerance allocation problem was formulated as a

probabilistic optimization problem (Lee and Woo, 1990, 1989) in

which the dimension and its corresponding tolerance were

represented as a random variable. The above optimization problem

was transformed into a deterministic optimization problem after using

the reliability index. The similar type of problems with nonlinear

nature was studied by Lee et. al (1993). Further, Chase et. al (1990)

solved the discrete tolerance synthesis problem after considering

substitute production processes for its components. Motivated by the

above mentioned works, many authors had worked for the solution of

optimal process and their sequences. The representative works

include Dong and Hu (1991), Dong and Soom (1991, 1990, 1989),

Dong et. al. (1994), Zhang et. al. (1992, 199a), Dong (1997b),

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Nagarwala et. al. (1994, 1995), Zhang and Wang (1993a,b,c, 1998),

and Roy and Fang (1997).

With the increase in the complexity of mechanical assemblies,

the tolerance allocation methods become impractical and a new area

of research was started in 1990. Taguchi et. al (1989) introduced the

concept of quality engineering, a discipline, in which the deviation

from the target value is minimum. Sometimes, it is also known as

robust engineering. A three-step procedure is followed in the product

design and manufacturing engineering phases to attain robustness.

Feng and Kusiak (1997) utilized both the quality loss and

manufacturing cost in the cost function. Moreover, Jeang (1995, 1997)

established a analytical model to attain product tolerances that

minimize the sum of manufacturing and quality loss cost. This was

obtained by not considering the production processes in the cost

function. Later on, a simultaneous tolerance synthesis method was

developed by Ye and Salustri (2004) after considering quality loss as a

continuous cost function.

Only linear tolerances were considered in all the STS models

and neglected the positional or geometrical tolerances. It was

understand that +/- system of dimensioning and tolerancing parts

was not sufficient every time to satisfy the design goal. Therefore, it is

necessary to consider the geometrical tolerances in the optimization

process. Geometrical Dimensioning & Tolerancing standard was

established by ANSI Y14.5-1994).

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Over the years, many methods were developed to establish

relationship between features and components in assembly. A GD&T

method of analyzing concentricity and run-out tolerances was

developed by Ngoi et al. (1997). Moreover, in the handbook of

Krulikowski (1992), the geometric stock method was discussed to

evaluate different kinds of stacks.

In the above mentioned works, the optimization of tolerance

allocation of mechanical assemblies was carried out using some

analytical methods. The solutions obtained by these methods may get

struck at the local minima. Thus, there is still a necessity to develop a

versatile and efficient algorithm to find near optimal tolerance values

for a mechanical assembly that reduces manufacturing and quality

loss cost.

2.3 EVOLUTIONARY ALGORITHMS BASED APPROACHES

In the recent past, evolutionary optimization approaches, like

Genetic algorithms (GA), Particle swarm optimization (PSO) and

Differential evolution (DE) have drawn a great deal of attention. These

approaches successfully avoid the local optima and thus substitute

the conventional gradient-based approaches (Tang and Wu, 2009).

The global optimal solution can be obtained more quickly with these

evolutionary optimization approaches through competition and

cooperation among the potential solutions that constitute the

population in the search space. Genetic Algorithm (GA), developed by

John Holland, is a global search and optimization procedure. This

algorithm works basically on the concept of “survival of the fittest”

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(Holland, 1992). Differential Evolution (DE) is a population based

stochastic, vector optimization process developed by Storn and Price,

(1997). The population of individuals in this approach is evolved using

search and selection. It is obtained by following a particular procedure

for constructing mutant vectors after utilizing the differences between

randomly selected vectors from the present population. Kennedy and

Eberhart (1995) developed Particle Swarm Optimization (PSO). It is an

optimization procedure based on swarm intelligence that was inspired

by the social nature of birds within a flock. The optimal value in this

approach was obtained by cooperation and sharing of information

between the individuals of a swarm (Kennedy and Eberhart, 1995).

The applicability of evolutionary algorithms to the tolerance

allocation problem had been demonstrated by very few researchers.

Ansary and Deiab (1997) solved the tolerance allocation problem for

simple assembly by using genetic algorithm. Whereas the same

problem was solved by Zhang and Wang (1993) after using simulated

annealing. Moreover, the same problem different stack-up conditions

was solved by Singh et al. (2003) by using genetic algorithm. The

above approaches had used only manufacturing cost as the objective

function. Gopala krishna and Mallikarjuna rao (2006) solved the

tolerance allocation problem by using scatter search algorithm. Later

on, pattern search algorithm had been used to solve the tolerance

allocation problem with asymmetric quality cost for a mechanical

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assembly by Sampath kumar et al. (2009). However, the authors had

not considered the geometric tolerances.

2.4 GAPS IN LITERATURE

The following gaps have been identified in the literature.

Most of the approaches developed for the tolerance allocation

use manufacturing cost only and very few have considered

quality cost also.

The existing tolerance allocation methods have not considered

positional tolerance. Moreover, the methods which are used to

convert geometrical tolerances into linear tolerances have also

not been attempted.

Most of the recent works considered only GA for optimization of

tolerance allocation for mechanical assemblies. Very few

researchers concentrated on the optimization of tolerance

allocation of mechanical assemblies utilizing other evolutionary

algorithms. No study attempted to identify the percentage

contribution of tolerances on total cost. Moreover, no systematic

study was there to compare the performance analysis of various

evolutionary algorithms.

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2.5 AIMS AND OBJECTIVES

After realizing the gaps in the literature, the aims and objective of

the present study have been set as follows:

The aim of the present study is to minimize the total cost (i.e.

manufacturing and quality cost) of the mechanical assemblies

after including the geometrical tolerance.

Three evolutionary algorithms, namely GA, DE and PSO have

been developed to find the near – optimal tolerance allocation

for the mechanical assemblies.

Various stack-up situations, like WC, RSS, SM and EMS

conditions are taken into account for the allocation of tolerances

to the mechanical assemblies.

Standard mechanical assemblies available in the literature (that

is, piston-cylinder assembly, wheel mount assembly and rotor

key assembly) are tested with the developed evolutionary

algorithms and their performances have also been compared.

Moreover, positional tolerance has also been considered in the

tolerance analysis of rotor-key assembly.

A systematic study is conducted to analyze the contribution of

individual tolerances on total cost.

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2.6 LAYOUT OF THE THESIS

The thesis has been organized in eight chapters. The contents of

Chapters 3 to 7 are given below.

Chapter 3: It includes mathematical formulation of the problem

related to the tolerance allocation of various mechanical assemblies.

Chapter 4: The problem of simultaneous tolerance allocation of piston-

cylinder, wheel-mounting and rotor-key assemblies is attempted as a

combined optimization problem using GA. The details of this study

were presented in this chapter.

Chapter 5: It elaborates the concept of Differential Evolution. It also

focuses on the constrained optimization problem of simultaneous

tolerance allocation of piston-cylinder, wheel-mounting and rotor-key

assemblies using DE.

Chapter 6: This chapter describes the basics of PSO and the

methodology followed in implementing PSO for the constrained

optimization problem of simultaneous tolerance allocation of above

mentioned mechanical assemblies.

Chapter 7: It gives the consolidated results of comparison on the

effectiveness of the developed approaches through computer

simulations.

Conclusions are drawn based on the work done and scope for

future work is indicated in the end.

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2.7 SUMMARY

This chapter provides with a survey of various analytical

methods for the tolerance allocation of mechanical assemblies, their

drawbacks have been identified. Evolutionary algorithms are used by

a several researchers to solve the said problem. After conducting a

critical literature review, the gaps in the literature have been identified

and based on the gaps, the aims and objective of the present thesis

have been decided. Finally, it provides with the layout of the thesis.